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Determination of pore size distribution of mesoporous materials by regularization
Studies in Surface Science and Catalysis 129 A. Sayari et al. (Editors) © 2000 Elsevier Science B.V. All rights reserved.
607
Determination of Pore Size Distribution of Mesoporous Materials by Regularization C. G. Sonwane and S. K. Bhatia* Department of Chemical Engineering, The University of Queensland, St. Lucia, Brisbane, QLD 4072, Australia
The development of regular mesoporous materials, MCM-41, has catalyzed considerable research in modeling adsorption phenomena at this scale. A new model for determining the pore size distribution of micro and mesoporous materials from gas adsorption isotherms is proposed here. The model uses the Dubinin-Rudushkevich (D-R) isotherm with Chen and Yang's correction in the micropore region. For the mesopore region, a recent model of the authors using a molecular-continuum approach for the multilayer region, and the Unilan model for the sub-monolayer region, has been extended. The experimental adsorption data is inverted using regularization to obtain the pore size distribution. The family of model mesoporous adsorbent, MCM-41, was chosen for testing the present model. The model was found to be successful in predicting the pore size distribution of pure as well as binary physical mixtures of MCM-41, with results in agreement with those from the XRD method. It was found that the BJH and the BdB methods under-predict while the Saito-Foley method over-predicts the pore diameter. The pore diameters obtained by the current model and the NLDFT were found to be close to actual pore diameters obtained by XRD. 1. INTRODUCTION Characterization of porous solids for determining structural parameters is important in catalysis, adsorption, separation and host-guest technologies. The important information obtained from the characterization of porous solids includes surface area, porosity and the pore size distribution. There are various techniques used for estimation of pore size distribution including gas adsorption, small angle X-ray as well as neutron scattering (SAXS and SANS), mercury porosimetry, nuclear magnetic resonance, thermoporometry, scanning as well as transmission electron microscopy (SEM and TEM) [1,2]. Each method has a limited length scale over which it is valid and useful for determining the pore size [3]. The recommendations of IPUAC for most of the methods are available [4]. For porous materials consisting of micropores (diameter< 2 nm) [1] and mesopores [2 nm < diameter < 50 nm], with a typical size range of 0.4-25 nm present in adsorbents and catalysts, nitrogen adsorption at its boiling point forms a convenient and inexpensive method of characterization that is widely used.
* To whom correspondence should be addressed. Email: sureshb(S:;cheque.uq.edu.au. Fax: +61 7 3365 4199, Telephone: +61 7 3365 4263.
608 The majority of the past studies of adsorption of vapors on mesoporous materials have focussed on the gathering of experimental data on adsorbents such as carbon, silica and alumina, while utilizing well established models such as the Kelvin, BJH, Saito-Foley (or HK for slit pores), Broekhoff-de Boer, Dubinin-Astakhov (DA) or others based on similar principle depending upon the pore size. Although simple and elegant, these approaches do involve approximations that are often unjustifiable. These models are either applicable explicitly to micropores or to the mesopores, and also they fail to explain complete pore structure of materials having both micro as well as mesopores. The difficulties in uniformly modeling the entire pore structure using adsorption isotherms arise because the adsorbate is in significantly different states in micropores and mesopores. Indeed, at low pressures at which the monolayer formation in the mesopores is not complete, the state of adsorbed molecules cannot be considered as that of the bulk phase. Consequently there is a need to have a model that is applicable over a wide range of pore sizes using adsorption isotherms of a variety of adsorbates and adsorbents. While this need is met by the newer molecular models and density functional theory approaches, they are computationally demanding and impractical for routine use. Recently [5] we have proposed a new hybrid isotherm for interpreting the adsorption of condensable gases on nonporous materials. This new hybrid isotherm incorporates the fluidsolid interaction potential within the framework of the classical approach in the multilayer region, while using existing models such as the Unilan for low pressures. The hybrid isotherm also satisfies the requirement of a Henry's law asymptote (depending upon the type of model used for the low-pressure region). The model was successftilly tested using isotherm data for nitrogen adsorption on nonporous silica, carbon and alumina, as well as benzene and hexane adsorption on nonporous carbon. Based on the data fits, out of several different alternative choices of model for the sub-monolayer region, the hybrid model involving the Freundlich and the Unilan models were found to be the most successful when combined with the multilayer model to predict the whole isotherm. The model can be easily modified and used for a cylindrical pore system. While the model for surface adsorption and subsequent condensation is well established, it is also recognized that in pores below a certain critical size (called micropores) these concepts do not hold and a different pore filling mechanism is operative. Although other models such as the Harvath-Kawazoe (slit geometry) [6] or the Saito-Foley (cylindrical) [7] exist that consider the micropore filling to be instantaneous, the Dubinin (Dubinin-Astakhav or Dubinin-Radushkevich) model is still more widely used [8]. At a given pressure (for low pressures) pores below a certain size will be completely filled by volume filling and the mesopores will have a submonolayer region. At higher pressures, in case of multilayer region, at a given pressure, pores with size smaller than a particular size will be filled by capillary condensate while the others will have a multilayer thickness of the adsorbate. This is depicted below in Figure 1. The isotherm models for micropores and mesopores can be used along with the equation [9]
0
to give a hybrid model for estimating the pore size distribution of the porous materials. Here Ca(P) is the total amount adsorbed, p(r,P)is the local effective density of the adsorbate in a pore of size r at pressure P. A key unknown in such a model is the critical micropore size below which this volume filling mechanism exists. lUPAC has recommended a limit of 2 nm
609 (diameter) as a standard (for both capillary condensation and the absence of hysteresis) [1], however recent studies with adsorption in regular mesoporous materials (MCM-41 type) suggest that the actual critical pore sizes vary and can be significantly different [10-13]. ii
. micropore adsorption,,^
\ /•
f(X)
/• • • •• /• * /• •
capillary /condensation
\
•v:
\
surface • adsorption
\
W
m
X.
^ X^(P)
X
Figure 1. Adsorption regimes in different pore size ranges. The adsorption of gases and vapors on mesoporous materials is generally characterized by multilayer adsorption followed by a distinct vertical step (capillary condensation) in the isotherm accompanied by a hysteresis loop. Studies of adsorption on MCM-41 have also demonstrated the absence of hysteresis for materials having pore size below a critical value. While this has been reported for silica gel and chromium oxide containing some mesopores, no consistent explanation has been offered [1]. However, conventional porous materials, having interconnected pores with a broader size distribution, are generally known to display a hysteresis loop with a point of closure which is characteristic of the adsorptive. These materials have an independent method of estimating the pore size from XRD and TEM, that allows comparison with theoretical results. Consequently, we have chosen these materials to test the proposed model. The family of these recently invented mesoporous materials, MCM-41, has attracted significant attention from a fundamental as well as applied perspective. They are considered as the most suitable model adsorbents currently available due to their array of uniform size pore channels (hexagonal/cylindrical pores) with negligible pore-networking or pore blocking effects. The prominent features of these materials include tunability of their pore diameter (in the range of 1.5-10 nm), a high surface area of 600-1300 m^/g, high thermal, hydrothermal and mechanical stability, ease of modification of the surface properties by incorporating heteroatoms such as Al, B, Ti, V and Mo as well as anchoring organic ligands, and their use as host materials for the construction of nano-structured materials by host-guest technology In this paper we have presented a new model for determining the pore size distribution of microporous and mesoporous materials. The model has been tested using the adsorption isotherms on pure as well as mixtures of MCM-41 materials. The experimental data of adsorption of nitrogen at 77.4 has been inverted using regularization technique. The results of PSD by the present model are compared with the pore size obtained from other classical methods, NLDFT [16] as well as the that obtained by X-ray diffraction methods.
610 2. THEORY 2.1 Micropores The fractional pore filling of the micropores of radius r at a given pressure P is given by the Dubinin-Radushkevich (DR) isotherm e{r, P) = exp[- {RT]n{PjP)//5EM'
J
(2)
where Rg is the ideal gas constant, T is temperature, EQ{r)is characteristic energy and P is similarity coefficient. Chen and Yang [17] have shown that the characteristic energy of adsorption is related to mean potential, C/> inside the pores by KN, = E,I3 (3) where No is the Avogadro's number and A^ is a constant. Here this approach can be used for cylindrical micropores using the Saito-Foley potential. The mean potential, O, for a cylindrical micropore system is given by [7] 4
dl
(4)
^ k+\
where NA and A^£ are number of oxygen atoms in the surface of zeohte and molecules/atoms in adsorbent respectively, do is the diameter of the adsorbent molecule and a* and fik are given by r(-4.5) (5) ' r{-4.5-k)r{k + \) „,/,__ r(-i.5) (6) r ( - i . 5 - ^ ) r ( ^ + i) Using Eqs. (3)-(6), the values of E {=PEo) are obtained in terms of the pore diameter as described by Chen and Yang [17]. 2.2 Mesopores In a recent article [5] we have proposed a new hybrid isotherm for adsorption in nonporous materials. The isotherm combines our recent molecular-continuum model for higher pressures, with other widely used models such as the Unilan model for the lowpressure region \-\-bPe' Q(P) = - ^ l n \ + bPe~ Is where Cm is the monolayer capacity and the constants are given by
-r^)J
C^b («• 2s _(l + ^.Fe~-^i\+bpe'
=
Pd$ /dt_
(7)
(8)
2sC„
e
p{e' -ke-' )
(9)
611 The hybrid isotherm uses our recent molecular-continuum model for higher pressures [18,19]. The model is extended here for adsorption in mesoporous materials. The equilibrium thickness t of the adsorbed layer at pressure P is given by ^ y vUr-t) c/>(t,r)^\v^dP = ^-^-^ ^ (10) Po (r-t-A/2/ where r is radius of the pore and (/)(t,r) is the position dependent incremental local potential due to the solid. The integral (second term) was obtained using the B WR equation of state. The fluid-solid interaction potential parameters were obtained by fitting the condensation pressures satisfying the stability boundary d^^_r^vJ(r-t + A/2) dt (r-t-X/lf
^^^^
to the MCM-41 data. The pressure at which the capillary evaporation occurs is given by ^^ ly vUr-t?(i>(r,r)^ \vdP-. 1^^ ^ = =0 (12) Po ' [(r-t)(r-t-A) + Acj,,/4] Consistent with Equation 1, with different adsorption regimes as shown in Figure 1, the integral can be split as r^
rp(p)
C„{P)= lp„X':P)f{>:P)dr+
r^^^^
\pXr.P)f{r.P)dr+
\ pXr.P)f{r.P)ir
(13)
permitting different forms of p{r, P) in each integral. Here p^ (r, P) represents the effective density for pores in which capillary condensation occurs, and p^ {r, P) that for pores in which multilayer surface coverage occurs. In the present case for the multilayer region
pc{'-'P)=r(
^—FTT7
\
2, n^^
^^^^
and for low pressure (15) with the effective density in the micropores given by pSr^P)=Pi expl- [RT\n{PjP)lPEXr)Y\
(16)
lUPAC defines the lower limit of mesopores as 2 nm [ 1 ] which was considered as the limit below which the adsorption will occur by volume filling. However, in our recent article, based on the tensile stress hypothesis, we have shown that this limit is different than lUPAC limit. Using the mechanical stability criterion for the cylindrical meniscus (during adsorption), the critical size is obtained from
P +r >
^
n?)
612 along with the thermodynamic stability condition (18)
{r-t-A/iy 3. RESULTS AND DISCUSSION
The above model was applied to our nitrogen isotherm data for different MCM-41 samples to extract pore size distributions (PSD's). Inversion of the adsorption integral was accomplished using the regularization package of Bhatia [10]. Figure 2 presents the PSD results for pure MCM-41 samples, along with the XRD estimates of the pore size. The details of the calculation of pore size from the XRD and gas adsorption have been given elsewhere [14,20]. Figure 3 depicts the results for a 1:1 (w/w) C12/C18 MCM-41 mixture.
30
40
50
60
pore diameter (A)
Figure 2. Comparison of pore size distribution of MCM-41 samples by regularization with the XRD pore diameter In the PSD calculations, LJ parameters of the silica-nitrogen (afs = 3.586 A and €fs/k = 66.6 K) and nitrogen-nitrogen ((afr=3.681 A, Gff/k=91.5 K) interactions were taken from our previous article [20]. The fluid was represented by the BWR equation of state. The properties of the adsorbate fluid (nitrogen) were taken as the bulk saturated liquid properties. For each isotherm, the values of the small regularization parameter were varied and the standard deviation, 5, was estimated. The final value was chosen in such a way that the standard deviation was constant and a stable non-negative PSD was obtained as described earher [10]. In the present model, the lower mesopore limit was taken as 2.4 nm which was estimated from our model describing stability of the adsorbate meniscus in mesopores. The D-R isotherm was applied for all the pores below 2.4 nm. Although the present calculations
613 show that the sample do not contain micropores, the applicability of the D-R equation up to 2.4 nm pore size, needs to be studied. Further, artifacts may arise at the transition between the different isotherms for the micropores and mesopores, but for the predominantly mesoporous MCM-41 this effect is not of significance. 0.15
30
40
30
50
40
50
pore diameter (A)
pore diameter (A)
0.10
E cv 0.05
0.00 30
40
pore diameter (A)
Figure 3. Comparison of pore size distribution of mixture of MCM-41 materials by regularization with the XRD pore diameter, as well as pore size distribution estimated from pure components: (a) C12+C18, (b)C12+C16,(c)C10+C14 The adsorption branch of the isotherm was used for the present calculations. It can be observed from Figure 2 that the predicted PSD is very close to the pore size from XRD except
614 for C8. This could be due to the random and disordered nature of the C8 sample, which was confirmed by HRTEM and fractal analysis in our previous work [20]. Consequently the XRD pore size of this material may not be meaningful. Mixtures were also prepared by physical mixing of different samples [12] and, as expected, showed peaks close to those of the pure components (c.f Figures 3 (a), (b) and (c)). This confirms that the present model provides a unique way of presenting the real pore structure of the system. The pore size distribution obtained by averaging the individual pure component PSD's of C12 and C18 is also shown in Figure 3 (a). Although the height of the peak for CI2 sample is higher by this averaging method, it should be noted that the area under the curve (pore volume) is the same. Similar results were obtained for the C12+C16 and C10+C14 mixtures, as shown in Figures 3 (b) and (c), with the result for the mixture PSD being very close to that of the average as well as the XRD method. Comparison of the pore size distribution determined by the present method with that from the classical methods such as the BJH, the Broekhoff-de Boer and the Saito-Foley methods is shown in Figure 4. Figure 5 shows a close resemblance of the results of our method with those from the recent NLDFT of Niemark et al. [16], and XRD pore diameter for their sample AMI. The results clearly indicate the utility of our method and accuracy comparable to the much more computationally demanding density functional theory. There are several other methods published recently (e. g. [21]), however space limitations do not permit comparison with these results here. It is hoped to discuss these in a future publication. 0.4
u.^o
. XRD
BJH fi 0.3 ]\
—
1 1 1
1 BdB 1 H 1 1 l|
0.20
p
Present theory
— NLDFT results
\ \
0.15 1 1 Ml
Q
5 0.2
Q
>
\\ present work
0.10
l\
1\ /\'^
0.1
'/ / // // // //
0.05
1V \ 0.0 1
^
20
, ,^A/.( 30
, ^
I' A \
40
, 1•
"
50
-^
60
PORE DIAMETER (A)
Figure 4. Comparison of pore size distribution of C18-MCM-41 sample determined by regularization with that from classical theories and the XRD pore diameter
0.00
yj 30
^^^r——.____ 40
50
PORE DIAMETER (A)
Figure 5. Comparison of pore size distribution of sample AMI of Niemark et al. [16] determined by current method with that from regularization with NLDFT, and the XRD pore diameter.
615 4. CONCLUSIONS A new model for determining the pore size distribution of micro and mesoporous materials from gas adsorption isotherm has been successfully proposed and tested. The present model was found to be successful in predicting the pore size distribution of pure as well as binary physical mixtures of MCM-41. 5. ACKNOWLEDGEMENTS The authors wish to acknowledge Mr. Russell Williams for his help with the computations. REFERENCES 1. Gregg, S.J. and Sing, K. S., Adsorption, Surface Area and Porosity, Academic Press, New York (1982). 2. Kaneko, K., J. Membrane Sci., 96 (1994) 59. 3. Sonwane, C. G. and Bhatia, S. K., Langmuir, 15 (1999) 2809. 4. Rouquerol, J., Avnir, D., Fairbridge, C. W., Evertt, D. H., Haynes, J. H., Pemicone, N., Ramsay, J. D., Sing, K. S. W. and Unger, K. K., Pure Appl. Chem., 66 (1994) 1739. 5. Sonwane, C. G. and Bhatia, S. K., submitted (1999). 6. Horvath, G. and Kawazoe, K., J. Chem. Eng. Jpn., 16 (1983) 470. 7. Saito, A. and Foley, H. C , AiChE J., 37 (1991) 429. 8. Do, D. D., Adsorption Analysis: Equilibria and Kinetics, Imperial College Press, London, (1998). 9. Bhatia, S. K., Chem. Eng. Sci., 53 (1998) 3239. 10. Inoue, S., Hanzawa, Y. and Kaneko, K., Langmuir, 14 (1998) 3079. 11. Morishige, K. and Shikimi, M., J. Chem. Phys., 108 (1998) 7821. 12. Sonwane, C. G. and Bhatia, S. K., Langmuir, 15 (1999) 5347. 13. Maddox, M. W., Olivier, J. P., Gubbins, K. E., Langmuir, 13 (1997) 1737. 14. Kruk, M., Jaroniec, M. and Sayari, A., J. Phys. Chem. B, 101 (1997) 583. 15. Sayari, A., Yang, Y., Kruk, M. and Jaroniec, M., J. Phys. Chem. B, 103 (1999) 3651. 16. Neimark, A. V., Ravikovitch, P. I. and Unger, K. K., J. Colloid Interface Sci., 207 (1998) 159. 17. Chen, S. G. and Yang, R. T., Langmuir, 10 (1994) 4244. 18. Bhatia, S. K. and Sonwane, C. G., Langmuir, 14 (1998) 1521. 19. Sonwane, C. G. and Bhatia, S. K., Chem. Eng. Sci., 53 (1998) 3143. 20. Sonwane, C. G., Bhatia, S. K , and Calos, N., Ind. Eng. Chem. Res., 37 (1998) 2271. 21. Kruk, M., Jaroniec, M. and Sayari, A., Langmuir, 13 (1997) 6267.
607
Determination of Pore Size Distribution of Mesoporous Materials by Regularization C. G. Sonwane and S. K. Bhatia* Department of Chemical Engineering, The University of Queensland, St. Lucia, Brisbane, QLD 4072, Australia
The development of regular mesoporous materials, MCM-41, has catalyzed considerable research in modeling adsorption phenomena at this scale. A new model for determining the pore size distribution of micro and mesoporous materials from gas adsorption isotherms is proposed here. The model uses the Dubinin-Rudushkevich (D-R) isotherm with Chen and Yang's correction in the micropore region. For the mesopore region, a recent model of the authors using a molecular-continuum approach for the multilayer region, and the Unilan model for the sub-monolayer region, has been extended. The experimental adsorption data is inverted using regularization to obtain the pore size distribution. The family of model mesoporous adsorbent, MCM-41, was chosen for testing the present model. The model was found to be successful in predicting the pore size distribution of pure as well as binary physical mixtures of MCM-41, with results in agreement with those from the XRD method. It was found that the BJH and the BdB methods under-predict while the Saito-Foley method over-predicts the pore diameter. The pore diameters obtained by the current model and the NLDFT were found to be close to actual pore diameters obtained by XRD. 1. INTRODUCTION Characterization of porous solids for determining structural parameters is important in catalysis, adsorption, separation and host-guest technologies. The important information obtained from the characterization of porous solids includes surface area, porosity and the pore size distribution. There are various techniques used for estimation of pore size distribution including gas adsorption, small angle X-ray as well as neutron scattering (SAXS and SANS), mercury porosimetry, nuclear magnetic resonance, thermoporometry, scanning as well as transmission electron microscopy (SEM and TEM) [1,2]. Each method has a limited length scale over which it is valid and useful for determining the pore size [3]. The recommendations of IPUAC for most of the methods are available [4]. For porous materials consisting of micropores (diameter< 2 nm) [1] and mesopores [2 nm < diameter < 50 nm], with a typical size range of 0.4-25 nm present in adsorbents and catalysts, nitrogen adsorption at its boiling point forms a convenient and inexpensive method of characterization that is widely used.
* To whom correspondence should be addressed. Email: sureshb(S:;cheque.uq.edu.au. Fax: +61 7 3365 4199, Telephone: +61 7 3365 4263.
608 The majority of the past studies of adsorption of vapors on mesoporous materials have focussed on the gathering of experimental data on adsorbents such as carbon, silica and alumina, while utilizing well established models such as the Kelvin, BJH, Saito-Foley (or HK for slit pores), Broekhoff-de Boer, Dubinin-Astakhov (DA) or others based on similar principle depending upon the pore size. Although simple and elegant, these approaches do involve approximations that are often unjustifiable. These models are either applicable explicitly to micropores or to the mesopores, and also they fail to explain complete pore structure of materials having both micro as well as mesopores. The difficulties in uniformly modeling the entire pore structure using adsorption isotherms arise because the adsorbate is in significantly different states in micropores and mesopores. Indeed, at low pressures at which the monolayer formation in the mesopores is not complete, the state of adsorbed molecules cannot be considered as that of the bulk phase. Consequently there is a need to have a model that is applicable over a wide range of pore sizes using adsorption isotherms of a variety of adsorbates and adsorbents. While this need is met by the newer molecular models and density functional theory approaches, they are computationally demanding and impractical for routine use. Recently [5] we have proposed a new hybrid isotherm for interpreting the adsorption of condensable gases on nonporous materials. This new hybrid isotherm incorporates the fluidsolid interaction potential within the framework of the classical approach in the multilayer region, while using existing models such as the Unilan for low pressures. The hybrid isotherm also satisfies the requirement of a Henry's law asymptote (depending upon the type of model used for the low-pressure region). The model was successftilly tested using isotherm data for nitrogen adsorption on nonporous silica, carbon and alumina, as well as benzene and hexane adsorption on nonporous carbon. Based on the data fits, out of several different alternative choices of model for the sub-monolayer region, the hybrid model involving the Freundlich and the Unilan models were found to be the most successful when combined with the multilayer model to predict the whole isotherm. The model can be easily modified and used for a cylindrical pore system. While the model for surface adsorption and subsequent condensation is well established, it is also recognized that in pores below a certain critical size (called micropores) these concepts do not hold and a different pore filling mechanism is operative. Although other models such as the Harvath-Kawazoe (slit geometry) [6] or the Saito-Foley (cylindrical) [7] exist that consider the micropore filling to be instantaneous, the Dubinin (Dubinin-Astakhav or Dubinin-Radushkevich) model is still more widely used [8]. At a given pressure (for low pressures) pores below a certain size will be completely filled by volume filling and the mesopores will have a submonolayer region. At higher pressures, in case of multilayer region, at a given pressure, pores with size smaller than a particular size will be filled by capillary condensate while the others will have a multilayer thickness of the adsorbate. This is depicted below in Figure 1. The isotherm models for micropores and mesopores can be used along with the equation [9]
0
to give a hybrid model for estimating the pore size distribution of the porous materials. Here Ca(P) is the total amount adsorbed, p(r,P)is the local effective density of the adsorbate in a pore of size r at pressure P. A key unknown in such a model is the critical micropore size below which this volume filling mechanism exists. lUPAC has recommended a limit of 2 nm
609 (diameter) as a standard (for both capillary condensation and the absence of hysteresis) [1], however recent studies with adsorption in regular mesoporous materials (MCM-41 type) suggest that the actual critical pore sizes vary and can be significantly different [10-13]. ii
. micropore adsorption,,^
\ /•
f(X)
/• • • •• /• * /• •
capillary /condensation
\
•v:
\
surface • adsorption
\
W
m
X.
^ X^(P)
X
Figure 1. Adsorption regimes in different pore size ranges. The adsorption of gases and vapors on mesoporous materials is generally characterized by multilayer adsorption followed by a distinct vertical step (capillary condensation) in the isotherm accompanied by a hysteresis loop. Studies of adsorption on MCM-41 have also demonstrated the absence of hysteresis for materials having pore size below a critical value. While this has been reported for silica gel and chromium oxide containing some mesopores, no consistent explanation has been offered [1]. However, conventional porous materials, having interconnected pores with a broader size distribution, are generally known to display a hysteresis loop with a point of closure which is characteristic of the adsorptive. These materials have an independent method of estimating the pore size from XRD and TEM, that allows comparison with theoretical results. Consequently, we have chosen these materials to test the proposed model. The family of these recently invented mesoporous materials, MCM-41, has attracted significant attention from a fundamental as well as applied perspective. They are considered as the most suitable model adsorbents currently available due to their array of uniform size pore channels (hexagonal/cylindrical pores) with negligible pore-networking or pore blocking effects. The prominent features of these materials include tunability of their pore diameter (in the range of 1.5-10 nm), a high surface area of 600-1300 m^/g, high thermal, hydrothermal and mechanical stability, ease of modification of the surface properties by incorporating heteroatoms such as Al, B, Ti, V and Mo as well as anchoring organic ligands, and their use as host materials for the construction of nano-structured materials by host-guest technology In this paper we have presented a new model for determining the pore size distribution of microporous and mesoporous materials. The model has been tested using the adsorption isotherms on pure as well as mixtures of MCM-41 materials. The experimental data of adsorption of nitrogen at 77.4 has been inverted using regularization technique. The results of PSD by the present model are compared with the pore size obtained from other classical methods, NLDFT [16] as well as the that obtained by X-ray diffraction methods.
610 2. THEORY 2.1 Micropores The fractional pore filling of the micropores of radius r at a given pressure P is given by the Dubinin-Radushkevich (DR) isotherm e{r, P) = exp[- {RT]n{PjP)//5EM'
J
(2)
where Rg is the ideal gas constant, T is temperature, EQ{r)is characteristic energy and P is similarity coefficient. Chen and Yang [17] have shown that the characteristic energy of adsorption is related to mean potential, C/> inside the pores by KN,
dl
(4)
^ k+\
where NA and A^£ are number of oxygen atoms in the surface of zeohte and molecules/atoms in adsorbent respectively, do is the diameter of the adsorbent molecule and a* and fik are given by r(-4.5) (5) ' r{-4.5-k)r{k + \) „,/,__ r(-i.5) (6) r ( - i . 5 - ^ ) r ( ^ + i) Using Eqs. (3)-(6), the values of E {=PEo) are obtained in terms of the pore diameter as described by Chen and Yang [17]. 2.2 Mesopores In a recent article [5] we have proposed a new hybrid isotherm for adsorption in nonporous materials. The isotherm combines our recent molecular-continuum model for higher pressures, with other widely used models such as the Unilan model for the lowpressure region \-\-bPe' Q(P) = - ^ l n \ + bPe~ Is where Cm is the monolayer capacity and the constants are given by
-r^)J
C^b («• 2s _(l + ^.Fe~-^i\+bpe'
=
Pd$ /dt_
(7)
(8)
2sC„
e
p{e' -ke-' )
(9)
611 The hybrid isotherm uses our recent molecular-continuum model for higher pressures [18,19]. The model is extended here for adsorption in mesoporous materials. The equilibrium thickness t of the adsorbed layer at pressure P is given by ^ y vUr-t) c/>(t,r)^\v^dP = ^-^-^ ^ (10) Po (r-t-A/2/ where r is radius of the pore and (/)(t,r) is the position dependent incremental local potential due to the solid. The integral (second term) was obtained using the B WR equation of state. The fluid-solid interaction potential parameters were obtained by fitting the condensation pressures satisfying the stability boundary d^^_r^vJ(r-t + A/2) dt (r-t-X/lf
^^^^
to the MCM-41 data. The pressure at which the capillary evaporation occurs is given by ^^ ly vUr-t?(i>(r,r)^ \vdP-. 1^^ ^ = =0 (12) Po ' [(r-t)(r-t-A) + Acj,,/4] Consistent with Equation 1, with different adsorption regimes as shown in Figure 1, the integral can be split as r^
rp(p)
C„{P)= lp„X':P)f{>:P)dr+
r^^^^
\pXr.P)f{r.P)dr+
\ pXr.P)f{r.P)ir
(13)
permitting different forms of p{r, P) in each integral. Here p^ (r, P) represents the effective density for pores in which capillary condensation occurs, and p^ {r, P) that for pores in which multilayer surface coverage occurs. In the present case for the multilayer region
pc{'-'P)=r(
^—FTT7
\
2, n^^
^^^^
and for low pressure (15) with the effective density in the micropores given by pSr^P)=Pi expl- [RT\n{PjP)lPEXr)Y\
(16)
lUPAC defines the lower limit of mesopores as 2 nm [ 1 ] which was considered as the limit below which the adsorption will occur by volume filling. However, in our recent article, based on the tensile stress hypothesis, we have shown that this limit is different than lUPAC limit. Using the mechanical stability criterion for the cylindrical meniscus (during adsorption), the critical size is obtained from
P +r >
^
n?)
612 along with the thermodynamic stability condition (18)
{r-t-A/iy 3. RESULTS AND DISCUSSION
The above model was applied to our nitrogen isotherm data for different MCM-41 samples to extract pore size distributions (PSD's). Inversion of the adsorption integral was accomplished using the regularization package of Bhatia [10]. Figure 2 presents the PSD results for pure MCM-41 samples, along with the XRD estimates of the pore size. The details of the calculation of pore size from the XRD and gas adsorption have been given elsewhere [14,20]. Figure 3 depicts the results for a 1:1 (w/w) C12/C18 MCM-41 mixture.
30
40
50
60
pore diameter (A)
Figure 2. Comparison of pore size distribution of MCM-41 samples by regularization with the XRD pore diameter In the PSD calculations, LJ parameters of the silica-nitrogen (afs = 3.586 A and €fs/k = 66.6 K) and nitrogen-nitrogen ((afr=3.681 A, Gff/k=91.5 K) interactions were taken from our previous article [20]. The fluid was represented by the BWR equation of state. The properties of the adsorbate fluid (nitrogen) were taken as the bulk saturated liquid properties. For each isotherm, the values of the small regularization parameter were varied and the standard deviation, 5, was estimated. The final value was chosen in such a way that the standard deviation was constant and a stable non-negative PSD was obtained as described earher [10]. In the present model, the lower mesopore limit was taken as 2.4 nm which was estimated from our model describing stability of the adsorbate meniscus in mesopores. The D-R isotherm was applied for all the pores below 2.4 nm. Although the present calculations
613 show that the sample do not contain micropores, the applicability of the D-R equation up to 2.4 nm pore size, needs to be studied. Further, artifacts may arise at the transition between the different isotherms for the micropores and mesopores, but for the predominantly mesoporous MCM-41 this effect is not of significance. 0.15
30
40
30
50
40
50
pore diameter (A)
pore diameter (A)
0.10
E cv 0.05
0.00 30
40
pore diameter (A)
Figure 3. Comparison of pore size distribution of mixture of MCM-41 materials by regularization with the XRD pore diameter, as well as pore size distribution estimated from pure components: (a) C12+C18, (b)C12+C16,(c)C10+C14 The adsorption branch of the isotherm was used for the present calculations. It can be observed from Figure 2 that the predicted PSD is very close to the pore size from XRD except
614 for C8. This could be due to the random and disordered nature of the C8 sample, which was confirmed by HRTEM and fractal analysis in our previous work [20]. Consequently the XRD pore size of this material may not be meaningful. Mixtures were also prepared by physical mixing of different samples [12] and, as expected, showed peaks close to those of the pure components (c.f Figures 3 (a), (b) and (c)). This confirms that the present model provides a unique way of presenting the real pore structure of the system. The pore size distribution obtained by averaging the individual pure component PSD's of C12 and C18 is also shown in Figure 3 (a). Although the height of the peak for CI2 sample is higher by this averaging method, it should be noted that the area under the curve (pore volume) is the same. Similar results were obtained for the C12+C16 and C10+C14 mixtures, as shown in Figures 3 (b) and (c), with the result for the mixture PSD being very close to that of the average as well as the XRD method. Comparison of the pore size distribution determined by the present method with that from the classical methods such as the BJH, the Broekhoff-de Boer and the Saito-Foley methods is shown in Figure 4. Figure 5 shows a close resemblance of the results of our method with those from the recent NLDFT of Niemark et al. [16], and XRD pore diameter for their sample AMI. The results clearly indicate the utility of our method and accuracy comparable to the much more computationally demanding density functional theory. There are several other methods published recently (e. g. [21]), however space limitations do not permit comparison with these results here. It is hoped to discuss these in a future publication. 0.4
u.^o
. XRD
BJH fi 0.3 ]\
—
1 1 1
1 BdB 1 H 1 1 l|
0.20
p
Present theory
— NLDFT results
\ \
0.15 1 1 Ml
Q
5 0.2
Q
>
\\ present work
0.10
l\
1\ /\'^
0.1
'/ / // // // //
0.05
1V \ 0.0 1
^
20
, ,^A/.( 30
, ^
I' A \
40
, 1•
"
50
-^
60
PORE DIAMETER (A)
Figure 4. Comparison of pore size distribution of C18-MCM-41 sample determined by regularization with that from classical theories and the XRD pore diameter
0.00
yj 30
^^^r——.____ 40
50
PORE DIAMETER (A)
Figure 5. Comparison of pore size distribution of sample AMI of Niemark et al. [16] determined by current method with that from regularization with NLDFT, and the XRD pore diameter.
615 4. CONCLUSIONS A new model for determining the pore size distribution of micro and mesoporous materials from gas adsorption isotherm has been successfully proposed and tested. The present model was found to be successful in predicting the pore size distribution of pure as well as binary physical mixtures of MCM-41. 5. ACKNOWLEDGEMENTS The authors wish to acknowledge Mr. Russell Williams for his help with the computations. REFERENCES 1. Gregg, S.J. and Sing, K. S., Adsorption, Surface Area and Porosity, Academic Press, New York (1982). 2. Kaneko, K., J. Membrane Sci., 96 (1994) 59. 3. Sonwane, C. G. and Bhatia, S. K., Langmuir, 15 (1999) 2809. 4. Rouquerol, J., Avnir, D., Fairbridge, C. W., Evertt, D. H., Haynes, J. H., Pemicone, N., Ramsay, J. D., Sing, K. S. W. and Unger, K. K., Pure Appl. Chem., 66 (1994) 1739. 5. Sonwane, C. G. and Bhatia, S. K., submitted (1999). 6. Horvath, G. and Kawazoe, K., J. Chem. Eng. Jpn., 16 (1983) 470. 7. Saito, A. and Foley, H. C , AiChE J., 37 (1991) 429. 8. Do, D. D., Adsorption Analysis: Equilibria and Kinetics, Imperial College Press, London, (1998). 9. Bhatia, S. K., Chem. Eng. Sci., 53 (1998) 3239. 10. Inoue, S., Hanzawa, Y. and Kaneko, K., Langmuir, 14 (1998) 3079. 11. Morishige, K. and Shikimi, M., J. Chem. Phys., 108 (1998) 7821. 12. Sonwane, C. G. and Bhatia, S. K., Langmuir, 15 (1999) 5347. 13. Maddox, M. W., Olivier, J. P., Gubbins, K. E., Langmuir, 13 (1997) 1737. 14. Kruk, M., Jaroniec, M. and Sayari, A., J. Phys. Chem. B, 101 (1997) 583. 15. Sayari, A., Yang, Y., Kruk, M. and Jaroniec, M., J. Phys. Chem. B, 103 (1999) 3651. 16. Neimark, A. V., Ravikovitch, P. I. and Unger, K. K., J. Colloid Interface Sci., 207 (1998) 159. 17. Chen, S. G. and Yang, R. T., Langmuir, 10 (1994) 4244. 18. Bhatia, S. K. and Sonwane, C. G., Langmuir, 14 (1998) 1521. 19. Sonwane, C. G. and Bhatia, S. K., Chem. Eng. Sci., 53 (1998) 3143. 20. Sonwane, C. G., Bhatia, S. K , and Calos, N., Ind. Eng. Chem. Res., 37 (1998) 2271. 21. Kruk, M., Jaroniec, M. and Sayari, A., Langmuir, 13 (1997) 6267.